Question
Each of these functions grows as x gets larger and larger. Which function eventually exceeds the others?
1. f (x) = 2x^2 - 9
2. g (x) = 2x + 8
3. h (x) = 2^x - 7
1. f (x) = 2x^2 - 9
2. g (x) = 2x + 8
3. h (x) = 2^x - 7
Answers
GPT-4o mini
To determine which function eventually exceeds the others as \( x \) gets larger, we can analyze the growth rates of the functions \( f(x) \), \( g(x) \), and \( h(x) \):
1. **Function \( f(x) = 2x^2 - 9 \)**:
- This is a quadratic function, and its leading term \( 2x^2 \) indicates that it grows at a rate proportional to \( x^2 \). As \( x \) becomes large, \( f(x) \) behaves like \( 2x^2 \).
2. **Function \( g(x) = 2x + 8 \)**:
- This is a linear function, and its growth rate is proportional to \( x \). As \( x \) becomes large, \( g(x) \) behaves like \( 2x \), which is significantly slower than quadratic growth.
3. **Function \( h(x) = 2^x - 7 \)**:
- This is an exponential function. The term \( 2^x \) suggests that it grows the fastest among the three. As \( x \) becomes very large, \( h(x) \) behaves like \( 2^x \), which outpaces both polynomial and linear growth.
### Comparison as \( x \) becomes large
- The quadratic function \( f(x) = 2x^2 - 9 \) will grow faster than the linear function \( g(x) = 2x + 8 \) for sufficiently large \( x \).
- However, the exponential function \( h(x) = 2^x - 7 \) grows much faster than the quadratic function \( f(x) \) as \( x \) increases since exponential functions eventually dominate polynomial functions.
### Conclusion
As \( x \) becomes larger and larger, the function \( h(x) = 2^x - 7 \) eventually exceeds both \( f(x) \) and \( g(x) \). Thus, the answer is:
**Function \( h(x) \) eventually exceeds the others.**
1. **Function \( f(x) = 2x^2 - 9 \)**:
- This is a quadratic function, and its leading term \( 2x^2 \) indicates that it grows at a rate proportional to \( x^2 \). As \( x \) becomes large, \( f(x) \) behaves like \( 2x^2 \).
2. **Function \( g(x) = 2x + 8 \)**:
- This is a linear function, and its growth rate is proportional to \( x \). As \( x \) becomes large, \( g(x) \) behaves like \( 2x \), which is significantly slower than quadratic growth.
3. **Function \( h(x) = 2^x - 7 \)**:
- This is an exponential function. The term \( 2^x \) suggests that it grows the fastest among the three. As \( x \) becomes very large, \( h(x) \) behaves like \( 2^x \), which outpaces both polynomial and linear growth.
### Comparison as \( x \) becomes large
- The quadratic function \( f(x) = 2x^2 - 9 \) will grow faster than the linear function \( g(x) = 2x + 8 \) for sufficiently large \( x \).
- However, the exponential function \( h(x) = 2^x - 7 \) grows much faster than the quadratic function \( f(x) \) as \( x \) increases since exponential functions eventually dominate polynomial functions.
### Conclusion
As \( x \) becomes larger and larger, the function \( h(x) = 2^x - 7 \) eventually exceeds both \( f(x) \) and \( g(x) \). Thus, the answer is:
**Function \( h(x) \) eventually exceeds the others.**